Regular colorings in regular graphs

Abstract

An (r − 1, 1)-coloring of an r-regular graph G is an edge coloring such that each vertex is incident to r − 1 edges of one color and 1 edge of a different color. In this paper, we completely characterize all 4-regular pseudographs (graphs that may contain parallel edges and loops) which do not have a (3, 1)-coloring. An {r − 1, 1}-factor of an r-regular graph is a spanning subgraph in which each vertex has degree either r − 1 or 1. We prove various conditions that that must hold for any vertex-minimal 5-regular pseudographs without (4, 1)-colorings or without {4, 1}-factors. Finally, for each r ≥ 6 we construct graphs that are not (r − 1, 1)-colorable and, more generally, are not (r − t, t)-colorable for small t.